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  • Plenary presentation for ATCM conference, Chang Mai, Thailand, December 2000

    Biological Brain, Mathematical Mind& Computational Computers

    (how the computer can support mathematical thinkingand learning)

    David TallMathematics Education Research Centre

    University of Warwick, UKe-mail:

    Brain, mind, computerBiological brain sensori-motor, visual, linguistic, process

    Mathematical Mind process, symbol, deduction, proof

    Computational computer algorithms, visual display

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    THE BIOLOGICAL BRAIN1990-2000 the decade of the brain.

    The Number Sense by Stanislas Dehaene (1997)The Mathematical Brain by Brian Butterworth (1999).

    Carter, R. (1998). Mapping the Mind. London: Weidenfeld &Nicholson.

    Crick, F. (1994). The Astonishing Hypothesis, London: Simon &Schuster.

    Pinker, S. (1997). How the Mind Works. New York: Norton.

    Edelman, G. M. (1992). Bright Air, Brilliant Fire, NY: BasicBooks, reprinted Penguin 1994.

    Greenfield, S. (1997). The Human Brain: A Guided Tour. London:Weidenfeld & Nicholson.

    Devlin. K. (2000), The Maths Gene: Why everyone has it, butmost people dont use it, London: Weidenfeld & Nicholson.

    Freeman, J. F. (1999) How the Brain makes up its Mind. Pheonix.

    Edelman, G. M. & Tononi, G. (2000). Consciousness: HowMatter Becomes Imagination. New York: Basic Books.

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    As a task to be learned is practiced, its performance becomesmore and more automatic; as this occurs, it fades fromconsciousness, the number of brain regions involved in the taskbecomes smaller. (Edelman & Tononi, 2000, p.51)

    Making complexity more manageable:The basic idea is that early processing is largely parallel alot of different activities proceed simultaneously. Then thereappear to be one or more stages where there is a bottleneck ininformation processing. Only one (or a few) object(s) canbe dealt with at a time. This is done by temporarily filteringout the information coming from the unattended objects. Theattentional system then moves fairly rapidly to the nextobject, and so on, so that attention is largely serial (i.e.,attending to one object after another) not highly parallel (as itwould be if the system attended to many things at once).

    (Crick, 1994, p. 61)

    This is made more efficient by making the consciouselements as small as possible, using words or symbols:

    I should also mention one other property of a symbolic system its compactibility a property that permits condensations of

    the order F = MA or s gt= 12 2, in each case the grammarbeing quite ordinary, though the semantic squeeze is quiteenormous.

    (Bruner, 1966, p. 12.)

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    MATHEMATICAL MINDConstructs to describe and explain the cognitive operationof the mathematical mind.

    the concept image, which refers to the total cognitivestructure in an individual mind associated with theconcept, including all mental pictures, associatedproperties and processes (Tall & Vinner, 1981),

    a theory of cognitive units (the mental chunks we use tothink with, and their related cognitive structure).(Barnard & Tall, 1997).

    A particular type of cognitive unit: the notion of procept, referring to the manner in which

    we cope with symbols representing both mathematicalprocesses and mathematical concepts . (Gray & Tall,1994). Examples include

    3+5, ax2+bx+c, d

    dxe xx sin( ), or xn


    n !=



    The brain is not configured for rapid & efficient arithmetic(Dehaene, 1997). It uses meaningful links betweencognitive units.

    Shaker Rasslan: a discussion about an algorithm fordivisibilty by seven, e.g. 121, 131, 119.

    Mathematical thinking is more than knowing proceduresto do. It involves a knowledge structure compatible withthe biological structure of the human brain:

    a huge store of knowledge and internal links, coping with many activities using a manageable focus

    of attention.

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    Consider, a linear relationship between two variables.This might be expressed in a variety of ways:

    an equation in the form y=mx+c, a linear relation Ax+By+C=0, a line through two given points, a line with given slope through a given point, a straight-line graph, a table of values, etc

    Successful students develop the idea of linear relationshipas a rich cognitive unit encompassing most of these links asa single entity.

    Less successful carry around a cognitive kit-bag ofisolated tricks to carry out specific algorithms. Short-termsuccess perhaps, long-term cognitive load and failure.

    Crowley (2000)

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    COMPUTATIONAL COMPUTERS complements human activity.

    The brain performs many activities simultaneously and isprone to error, concept image as support.

    the computer carries out individual algorithms accuratelyand with great speed. No conceptual baggage.

    Arithmetic algorithms for four rules +, , , ( etc)Algebra rules for substitution, eg X*(Y) may be

    replaced by (X*Y) where X and Y are expressions.Derive (an early version) simplified

    n n(x+h) -x--------- h

    to give

    n n(x+h) x----- - --- . h h

    the limit option as h tends to 0, gave not nxn-1, but n LN(x)-LN(x/n) .

    The current version of Derive has added a new rule to give:

    n1n x .




    x0 calculated mentally using graphic image.

    Computer algebra system may use:


    lim(sin )

    ( )cos

    x x



    D x

    D x = = =

    0 0



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    Given the constraints and support in the biological brain,the concept imagery in the mathematical mind can be verydifferent from the working of the computational computer.A professional mathematician with mathematical cognitiveunits may use the computer in a very different way from thestudent who is meeting new ideas in a computer context.

    Students using Derive on hand-held computers to drawgraphs of functions were asked:

    What can you say about u if u=v+3, and v=1?

    None of seventeen students improved from pre-test to post-test and six successful on the pre-test failed on the post-test.

    Hunter, Monaghan & Roper (1993)

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    A symbol manipulator replaces the mathematicalprocedures of differentiation by the selection of a sequenceof procedures in the software. For instance Derive requiresthe user to take the following sequence of decisions carriedout by touching the appropriate keys:

    select Author and type in the expression, select Calculus, then Derivative, specify the variable (e.g. x), Simplify the result.

    A comparison of two schools in the UK, one following astandard course, one using Derive is as follows:

    Please explain the meaning of lim( ) ( )


    f x h f x

    h+ -

    0 .

    All the students in school A gave satisfactory theoreticalexplanations of the expression but none gave any examples.However, none of the Derive group gave theoreticalexplanations and only two students [out of seven] mentionedthe words gradient or differentiate. Four of the Derive groupgave examples. They replaced f(x) with a polynomial andperformed or described the sequence of key strokes to calculatethe limit. (Monaghan, Sun & Tall, 1994.)

    Students using the symbol manipulator saw differentiationas a sequence of keystrokes in a specific symbolic examplerather than a conceptual idea of rate of change.

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    COGNITIVE DEVELOPMENT OF SYMBOLS(1) arithmetic procepts, 5+4, 3 4, 12 2 3+ ,

    15423,computational as processes and concepts. in the sum 8+6,

    the concept 6 can be linked to the operation 2+4, which canbe combined in the sum 8+2+4 to give 10+4 which is 14.

    (2) algebraic procepts, e.g 2+3x, only a potentialprocess (of numerical substitution), yet expressions are treated asmanipulable concepts using usual algebraic rules.

    Meaningful power operations such as

    2 2 2 2 2 2 2 23 2 5 = =( ) ( )act as a cognitive basis for the power law

    x x xm n m n = +(3) implicit procepts, x

    12 , x0 , x-1 , where original

    meaning of xn no longer applies, but is implicit in thegeneralised power law.

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    (4) limit procepts,

    e.g. limx a

    x a

    x a--

    3 3

    , 1

    21 nn=

    .These have potentially infinite processes getting close to a

    limit value, but this may not be computable in a finite number ofcomputations. Limits are often conceived as variable quantitieswhich get arbitrarily close to a limiting value, rather than the limitvalue itself.

    (5) calculus procepts,

    e.g. d x e


    x( )2

    , sin cosmx nx dx0p

    These (may) have finite operational algorithms of computation

    (using various rules for differentiation and integration).

    Should the limit procept be the first idea in calculus?

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    Formal definitionsand proof

    (dynamic limit concept)


    (implicit algebraic concepts)



    defined properties (axioms and definitions)logical processes (deductions)formally constructed concepts (theorems)

    computational processes (rules)manipulable concepts (formulas)

    potentially infinite processesarbitrarily small, close, or large concepts

    (variable quantities)

    potential processes (evaluating expressions)manipulable concepts (expressions)

    implicit processes (generalised properties)manipulable concepts (generalised expressions)

    computational processes (arithmetic)computational concepts (numbers)

    Changing meaning of symbols in arithmetic,algebra and calcul


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